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| The process of transforming ''b'' into ''B'' is simply the subtraction of all components of ''a'' from ''b''. This is a linear combination and does not change the column space of a system that includes both ''a'' and ''b''. [[LinearAlgebra/Projections#Vectors|Projections]] are a complimentary idea; ''p'' is the component of ''a'' that estimates ''b''. The process of '''orthonormalization''' is the same as computing projections but the error term ''e'' is the desired result. Recall that ''e = b - ax̂'' and ''x̂ = (A^T^b)/(A^T^A)''. Therefore, ''B = b - A (A^T^b)/(A^T^A)''. | The process of transforming ''b'' into ''B'' is simply the subtraction of all components of ''a'' from ''b''. This is a linear combination and does not change the column space of a system that includes both ''a'' and ''b''. [[LinearAlgebra/Projection|Projections]] are a complimentary idea; ''p'' is the component of ''a'' that estimates ''b''. The process of '''orthonormalization''' is the same as computing projections but the error term ''e'' is the desired result. Recall that ''e = b - ax̂'' and ''x̂ = (A^T^b)/(A^T^A)''. Therefore, ''B = b - A (A^T^b)/(A^T^A)''. |
Orthonormalization
Gram-Schmidt orthonormalization is a process for making vectors into orthonormal vectors.
Contents
Vectors
Two vectors a and b can be orthonormalized into A and B.
Orthogonality is a property of two vectors, not one. Therefore a needs no transformation and becomes A.
The process of transforming b into B is simply the subtraction of all components of a from b. This is a linear combination and does not change the column space of a system that includes both a and b. Projections are a complimentary idea; p is the component of a that estimates b. The process of orthonormalization is the same as computing projections but the error term e is the desired result. Recall that e = b - ax̂ and x̂ = (ATb)/(ATA). Therefore, B = b - A (ATb)/(ATA).
To transform another vector c into being orthogonal to both A and B, apply the same process for each component: C = c - A (ATc)/(ATA) - B (BTc)/(BTB).
The orthogonal vectors are then normalized by scaling to their Euclidean distances, as A/||A|| and B/||B||.
Matrices
The process applied to vectors is also applicable to the columns in a matrix. Instead of vectors a and b, use v1 and v2 in V. The process yields u1 and u2 in U. Then the columns are normalized into Q like q1 = u1/||u1||.
To re-emphasize, this is a linear combination generalized as A = QR, and does not change the column space of A.
Note that Q is a matrix with orthonormal columns; it must also be square to be called an orthogonal matrix.